1

Chapter 2: Time Value of Money

Future value Present value Rates of return Amortization

2

Time lines show timing of cash flows.

CF0 CF1 CF3CF2

0 1 2 3i%

Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

3

Time line for a $100 lump sum due at the end of Year 2.

100

0 1 2 Yeari%

4

Time line for an ordinary annuity of $100 for 3 years

100 100100

0 1 2 3i%

5

Time line for uneven CFs

100 50 75

0 1 2 3i%

-50

6

FV of an initial $100 after3 years (i = 10%)

FV = ?

0 1 2 310%

Finding FVs (moving to the righton a time line) is called compounding.

100

7

After 2 years

FV1 = PV + INT1 = PV + PV (i)= PV(1 + i)= $100(1.10)= $110.00.

FV2 = FV1(1+i) = PV(1 + i)(1+i)= PV(1+i)2

= $100(1.10)2

= $121.00.

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After 3 years

FV3 = FV2(1+i)=PV(1 + i)2(1+i)= PV(1+i)3

= $100(1.10)3

= $133.10

In general,FVn = PV(1 + i)n.

9

Three Ways to Find FVs

Solve the equation with a regular calculator.

Use a financial calculator. Use a spreadsheet.

10

Financial calculator: HP10BII

Adjust display brightness: hold down ON and push + or -.

Set number of decimal places to display: Orange Shift key, then DISP key (in orange), then desired decimal places (e.g., 3).

To temporarily show all digits, hit Orange Shift key, then DISP, then =

11

HP10BII (Continued)

To permanently show all digits, hit ORANGE shift, then DISP, then . (period key)

Set decimal mode: Hit ORANGE shift, then ./, key. Note: many non-US countries reverse the US use of decimals and commas when writing a number.

12

HP10BII: Set Time Value Parameters

To set END (for cash flows occurring at the end of the year), hit ORANGE shift key, then BEG/END.

To set 1 payment per period, hit 1, then ORANGE shift key, then P/YR

13

Financial calculators solve this equation:

FVn + PV (1+i)n = 0.

There are 4 variables. If 3 are known, the calculator will solve for the 4th.

Financial Calculator Solution

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3 10 -100 0N I/YR PV PMT FV

133.10

Clearing automatically sets everything to 0, but for safety enter PMT = 0.

Set: P/YR = 1, END.

INPUTS

OUTPUT

Here’s the setup to find FV

15

Spreadsheet Solution

Use the FV function: see spreadsheet in “FM11 Ch 02 Mini Case.xls”

= FV(Rate, Nper, Pmt, PV) = FV(0.10, 3, 0, -100) = 133.10

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10%

What’s the PV of $100 due in 3 years if i = 10%?

Finding PVs is discounting, and it’s the reverse of compounding.

100

0 1 2 3

PV = ?

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Solve FVn = PV(1 + i )n for PV

PV =

FVn

(1 + i)n

= FVn( )1

1 + i

n

PV = $100

11.10

= $100 0.7513 = $75.13.

3

18

3 10 0 100N I/YR PV PMT FV

-75.13

Either PV or FV must be negative. HerePV = -75.13. Put in $75.13 today, take out $100 after 3 years.

INPUTS

OUTPUT

Financial Calculator Solution

19

Spreadsheet Solution

Use the PV function: see spreadsheet in “FM11 Ch 02 Mini Case.xls”

= PV(Rate, Nper, Pmt, FV

= PV(0.10, 3, 0, 100) = -75.13

20

20%

2

0 1 2 ?

-1 FV = PV(1 + i)n

$2 = $1(1 + 0.20)n

(1.2)n = $2/$1 = 2nLN(1.2) = LN(2) n = LN(2)/LN(1.2) n = 0.693/0.182 = 3.8.

Finding the Time to Double

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20 -1 0 2N I/YR PV PMT FV

3.8

INPUTS

OUTPUT

Financial Calculator Solution

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Spreadsheet Solution

Use the NPER function: see spreadsheet in “FM11 Ch 02 Mini Case.xls”

= NPER(Rate, Pmt, PV, FV)

= NPER(0.10, 0, -1, 2) = 3.8

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?%

2

0 1 2 3

-1 FV = PV(1 + i)n

$2 = $1(1 + i)3

(2)(1/3) = (1 + i) 1.2599 = (1 + i) i = 0.2599 = 25.99%.

Finding the Interest Rate

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3 -1 0 2N I/YR PV PMT FV

25.99

INPUTS

OUTPUT

Financial Calculator

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Spreadsheet Solution

Use the RATE function:

= RATE(Nper, Pmt, PV, FV)

= RATE(3, 0, -1, 2) = 0.2599

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Ordinary Annuity

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3i%

PMT

Annuity Due

Ordinary Annuity vs. Annuity Due

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What’s the FV of a 3-year ordinary annuity of $100 at 10%?

100 100100

0 1 2 310%

110 121FV = 331

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FV Annuity Formula

The future value of an annuity with n periods and an interest rate of i can be found with the following formula:

= PMT

(1+i)n-1

i

= 100

(1+0.10)3-1

0.10= 331

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Financial Calculator Formula for Annuities

Financial calculators solve this equation:

FVn + PV(1+i)n + PMT

(1+i)n-1

i= 0.

There are 5 variables. If 4 are known, the calculator will solve for the 5th.

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3 10 0 -100

331.00N I/YR PV PMT FV

Have payments but no lump sum PV, so enter 0 for present value.

INPUTS

OUTPUT

Financial Calculator Solution

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Spreadsheet Solution

Use the FV function: see spreadsheet.

= FV(Rate, Nper, Pmt, Pv) = FV(0.10, 3, -100, 0) = 331.00

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What’s the PV of this ordinary annuity?

100 100100

0 1 2 310%

90.91

82.64

75.13248.69 = PV

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PV Annuity Formula

The present value of an annuity with n periods and an interest rate of i can be found with the following formula:

= PMT1 -

1

(1+i)n

i

= 100

1 - 1

(1+0.10)3

0.10

= 248.69

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Have payments but no lump sum FV, so enter 0 for future value.

3 10 100 0N I/YR PV PMT FV

-248.69

INPUTS

OUTPUT

Financial Calculator Solution

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Spreadsheet Solution

Use the PV function: see spreadsheet.

= PV(Rate, Nper, Pmt, Fv) = PV(0.10, 3, 100, 0) = -248.69

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Find the FV and PV if theannuity were an annuity due.

100 100

0 1 2 3

10%

100

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PV and FV of Annuity Due vs. Ordinary Annuity

PV of annuity due:= (PV of ordinary annuity) (1+i) = (248.69) (1+ 0.10) = 273.56

FV of annuity due:= (FV of ordinary annuity) (1+i)= (331.00) (1+ 0.10) = 364.1

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3 10 100 0

-273.55 N I/YR PV PMT FV

INPUTS

OUTPUT

PV of Annuity Due: Switch from “End” to “Begin

Then enter PV = 0 and press FV to findFV = $364.10.

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Excel Function for Annuities Due Change the formula to:

=PV(10%,3,-100,0,1)

The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due:

=FV(10%,3,-100,0,1)

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What is the PV of this uneven cash flow stream?

0

100

1

300

2

300

310%

-50

4

90.91247.93225.39-34.15

530.08 = PV

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Financial calculator: HP10BII

Clear all: Orange Shift key, then C All key (in orange).

Enter number, then hit the CFj key. Repeat for all cash flows, in order. To find NPV: Enter interest rate

(I/YR). Then Orange Shift key, then NPV key (in orange).

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Financial calculator: HP10BII (more)

To see current cash flow in list, hit RCL CFj CFj

To see previous CF, hit RCL CFj – To see subsequent CF, hit RCL CFj

+ To see CF 0-9, hit RCL CFj 1 (to see

CF 1). To see CF 10-14, hit RCL CFj . (period) 1 (to see CF 11).

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Input in “CFLO” register: CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50

Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV.)

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Excel Formula in cell A3: =NPV(10%,B2:E2)

A B C D E

1 0 1 2 3 4

2 100 300 300 -50

3 530.09

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Nominal rate (iNom)

Stated in contracts, and quoted by banks and brokers.

Not used in calculations or shown on time lines

Periods per year (m) must be given. Examples:

8%; Quarterly 8%, Daily interest (365 days)

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Periodic rate (iPer )

iPer = iNom/m, where m is number of

compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.

Used in calculations, shown on time lines. Examples:

8% quarterly: iPer = 8%/4 = 2%.

8% daily (365): iPer = 8%/365 = 0.021918%.

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Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant?

LARGER!

If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

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FV with Different Compounding Periods (e.g., $100 at a 12% nominal rate with semiannual compounding for 5 years)

= $100(1.06)10 = $179.08.

FV = PV 1 .+ i

mn

Nommn

FV = $100 1 + 0.12

25S

2x5

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FV of $100 at a 12% nominal rate for 5 years with different compounding

FV(Annual)= $100(1.12)5 = $176.23.FV(Semiannual)= $100(1.06)10=$179.08.FV(Quarterly)= $100(1.03)20 = $180.61.FV(Monthly)= $100(1.01)60 = $181.67.FV(Daily) = $100(1+(0.12/365))(5x365) = $182.19.

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Effective Annual Rate (EAR = EFF%) The EAR is the annual rate which causes

PV to grow to the same FV as under multi-period compounding. Example: Invest $1 for one year at 12%, semiannual:

FV = PV(1 + iNom/m)m FV = $1 (1.06)2 = 1.1236.

EFF% = 12.36%, because $1 invested for one year at 12% semiannual compounding would grow to the same value as $1 invested for one year at 12.36% annual compounding.

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Comparing Rates

An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons.

Banks say “interest paid daily.” Same as compounded daily.

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EFF% = - 1(1 + )iNom

m

m

= - 1.0(1 + )0.122

2

= (1.06)2 - 1.0 = 0.1236 = 12.36%.

EFF% for a nominal rate of 12%, compounded semiannually

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Finding EFF with HP10BII

Type in nominal rate, then Orange Shift key, then NOM% key (in orange).

Type in number of periods, then Orange Shift key, then P/YR key (in orange).

To find effective rate, hit Orange Shift key, then EFF% key (in orange).

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EAR (or EFF%) for a Nominal Rate of of 12%

EARAnnual = 12%.

EARQ = (1 + 0.12/4)4 - 1 = 12.55%.

EARM = (1 + 0.12/12)12 - 1 = 12.68%.

EARD(365) = (1 + 0.12/365)365 - 1 = 12.75%.

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Can the effective rate ever be equal to the nominal rate?

Yes, but only if annual compounding is used, i.e., if m = 1.

If m > 1, EFF% will always be greater than the nominal rate.

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When is each rate used?

iNom: Written into contracts, quoted by banks and brokers. Not used in calculations or shownon time lines.

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iPer: Used in calculations, shown on time lines.

If iNom has annual compounding,then iPer = iNom/1 = iNom.

When is each rate used? (Continued)

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When is each rate used? (Continued)

EAR (or EFF%): Used to compare returns on investments with different payments per year.

Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.

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Amortization

Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

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Step 1: Find the required payments.

PMT PMTPMT

0 1 2 310%

-1,000

3 10 -1000 0

INPUTS

OUTPUT

N I/YR PV FVPMT

402.11

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Find interest charge and principal repayment for Year 1.

Step 1: Find interestINTt = Beg balt (i)INT1 = $1,000(0.10) = $100.

Step 2: Find repaid princ.Repmt = PMT - INT

= $402.11 - $100 = $302.11.

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Step 4: Find ending balance after Year 1.

End bal = Beg bal - Repmt= $1,000 - $302.11 = $697.89.

Repeat these steps for Years 2 and 3to complete the amortization table.

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Amortization Table

YEARBEG BAL PMT INT

PRIN PMT

END BAL

1 $1,000

$402 $100 $302 $698

2 698 402 70 332 366

3 366 402 37 366 0

TOT 1,206.34 206.34 1,000

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Interest declines because outstanding balance declines.

$0$50

$100$150$200$250$300$350$400$450

PMT 1 PMT 2 PMT 3

Interest

Principal

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Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, and more. They are very important!

Financial calculators (and spreadsheets) are great for setting up amortization tables.

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Fractional Time Periods

On January 1 you deposit $100 in an account that pays a nominal interest rate of 11.33463%, with daily compounding (365 days).

How much will you have on October 1, or after 9 months (273 days)? (Days given.)

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iPer = 11.33463%/365= 0.031054% per day.

FV=?

0 1 2 273

0.031054%

-100

Convert interest to daily rate

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273 -100 0

108.85

INPUTS

OUTPUT

N I/YR PV FVPMT

iPer = iNom/m= 11.33463/365= 0.031054% per day.

Calculator Solution

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What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually?

0 1

100

2 35%

4 5 6 6-mos. periods

100 100

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Non-matching rates and periods

Payments occur annually, but compounding occurs each 6 months.

So we can’t use normal annuity valuation techniques.

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1st Method: Compound Each CF

0 1

100

2 35%

4 5 6

100 100.00110.25121.55331.80

FVA3 = $100(1.05)4 + $100(1.05)2 + $100= $331.80.

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2nd Method: Treat as an annuity, use financial calculator

Find the EAR for the quoted rate:

EAR = (1 + ) - 1 = 10.25%. 0.10

22

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3 10.25 0 -100

INPUTS

OUTPUT N I/YR PV FVPMT

331.80

Use EAR = 10.25% as the annual rate in calculator.

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What’s the PV of this stream?

0

100

15%

2 3

100 100

90.7082.2774.62

247.59

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Comparing Investments You are offered a note which pays $1,000

in 15 months (or 456 days) for $850. You have $850 in a bank which pays a 6.76649% nominal rate, with 365 daily compounding, which is a daily rate of 0.018538% and an EAR of 7.0%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless.

Should you buy it?

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iPer = 0.018538% per day.

1,000

0 365 456 days

-850

Daily time line

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Method 1: Greatest Future Wealth

Find FV of $850 left in bank for15 months and compare withnote’s FV = $1,000.FVBank = $850(1.00018538)456

= $924.97 in bank.Buy the note: $1,000 > $924.97.

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456 -850 0

924.97

INPUTS

OUTPUT

N I/YR PV FVPMT

iPer = iNom/m= 6.76649%/365= 0.018538% per day.

Calculator Solution to FV

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Find PV of note, and comparewith its $850 cost:PV = $1,000/(1.00018538)456

= $918.95.

Method 2: Greatest Present Wealth

80

456 .018538 0 1000

-918.95

INPUTS

OUTPUT

N I/YR PV FVPMT

6.76649/365 =

PV of note is greater than its $850 cost, so buy the note. Raises your wealth.

Financial Calculator Solution

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Find the EFF% on note and compare with 7.0% bank pays, which is your opportunity cost of capital:

FVn = PV(1 + i)n

$1,000 = $850(1 + i)456

Now we must solve for i.

Method 3: Rate of Return

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456 -850 0 1000

0.035646% per day

INPUTS

OUTPUT

N I/YR PV FVPMT

Convert % to decimal:Decimal = 0.035646/100 = 0.00035646.EAR = EFF% = (1.00035646)365 - 1 = 13.89%.

Calculator Solution

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P/YR = 365NOM% = 0.035646(365) = 13.01 EFF% = 13.89Since 13.89% > 7.0% opportunity cost,buy the note.

Using Interest Conversion